Reinforced Concrete Beam-Column Joint: Macroscopic

Super-element models

-Nilanjan Mitra

(work performed as a PhD student while at University of Washington between 2001-2006)

Need for the study

Reinforced concrete beam column jointssubjected to earthquake loading

Experimental Investigation@ UW

I-280 Freeway, San Francisco, CAfollowing Loma Prieta Earthquake in 1989

Courtesy: NISEE, Univ. of California, Berkeley.

Loading in a joint region

Earthquake Loading of Beam-Column Joint

compression resultant (concrete and steel)

shear resultant (concrete)

Earthquake Loading of Beam-Column Joint

compression resultant (concrete and steel)

compression resultant (concrete and steel)

shear resultant (concrete)

shear resultant (concrete) tension resultant (steel)

anchorage bond stress acting on joint core concrete

compression force carried by joint core concrete

Internal load distribution in a joint

Macroscopic beam-column joint element models

Macroscopic beam-column joint element models

Macroscopic beam-column joint element models

shear panel

external node

internal node

rigid externalinterface plane

shown with finite widthto facilitate discussion

beam element

zero-width region

interface-shear spring

bar-slip spring

zero-length

zero-length

elem

ent

colu

mn

Proposed Beam-column super-element model 4-noded 12-dof element 8 bar-slip springs to simulate

anchorage failure 4 interface-shear springs to simulate

shear transfer failure at joint interface 1 shear-panel to simulate inelastic

action of shear within joint core

Note: The location of the bar-slipsprings is at the centroid of thetension-compression couple at nominalstrength of the beams.

[Mitra & Lowes; J. Structural Eng. ASCE, 2007: 133 (1): 105-20 ]

Joint element formulation: Kinematics

External, Internal and Component deformation

Joint element formulation: Equilibrium

External, Internal and Component forces

Solution of element state achieved by an iterative procedure and requires solving for zero reaction in the 4 internal degrees of freedom

Characterized by Response envelope Unload reload path Damage rules

Hysteretic one dimensional material model

deformation

load

(ePd1,ePf1)

(ePd4,ePf4)

(ePd3,ePf3)(ePd2,ePf2)

(eNd3,eNf3)(eNd2,eNf2)

(*,uForceP.ePf3)

(dmin,f(dmin))

(dmax,f(dmax))

(rDispP.dmax,rForceP.f(dmax))

(rDispN.dmin,rForceN.f(dmin))

(*,uForceN.eNf3)

(eNd1,eNf1)

(eNd4,eNf4)

Damage simulation in material model

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-8

-6

-4

-2

0

2

4

6

8

deformation

load

without damagewith unloading stiffness damage

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-8

-6

-4

-2

0

2

4

6

8

deformationlo

ad

without damagewith reloading stiffness damage

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-8

-6

-4

-2

0

2

4

6

8

deformation

load

without damagewith strength damage

( ) ( )( )3 41 max 2i d = +

max minmax

max min

max. ,i id d

ddef def

=

( ).f No of load cycles =

( )f Accumulated Energy =

( )0 1 ki ik k = ( ) ( ) ( )max max 0 1

fii

f f = ( ) ( ) ( )max max 0 1

dii

d d = +

Damage simulation in material model

load historyi

monotonic

monotonic load history

dEE

EgE dE

= =

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-8

-6

-4

-2

0

2

4

6

8

deformation

load

without damagewith all 3 damage rules

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-5

-4

-3

-2

-1

0

1

2

3

4

5

deformation

load

with all 3 damages (Energy)with all 3 damages (Cyclic)

max4duu

=

Energy criterion

No. of load cycle criterion: rain-flow-counting algorithm

Shear-panel calibration

column

shear panel

Shear panel envelope calibration MCFT Diagonal compression strut

Compression envelope reduction

Determination of hysteretic model parameters

Typical response envelope

Observed Simulated

Spec

imen

SE8

(S

teve

ns e

t al.

1987

)

-0.012 -0.008 -0.004 0 0.004 0.008 0.012-10

-8

-6

-4

-2

0

2

4

6

8

10

Shear strain

Shea

r st

ress

(MPa

)

Shear panel envelope calibration using proposedDiagonal compression strut mechanism

_ cosc strut strut strutstrut

jnt

f ww

=

Mander et al. (1988) concrete

Column longitudinal and joint hoop

steel confine the strut.

Reduction in concrete to account for

perpendicular tensile stress to the strut

cyclic loading.

Strut force is converted to panel shear

stress as

2_

_3.62 2.82 1 0.39

0.45 0.39

c strut t t t

c Mander cc cc cc

t

cc

ff

= + 0

Data with j = 0

Vecchio 1986Stevens 1991Hsu 1995Noguchi 1992Proposed eq. for

j > 0

Proposed eq. for j = 0

2_

_0.36 0.60 1 0.83

0.75 0.83

c strut t t t

c Mander cc cc cc

t

cc

ff

= + Eq. for Eq. for

Comparison of MCFT and Diagonal Compression Strut model in shear-panel envelope calibration

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

mcf

t_cy

clic

/ m

ax

0.55JFBYJFBY

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

dia

gona

l_st

rut /

m

ax

JFBYJFBY

[Lowes, Altoontash and Mitra, J. Structural Eng. ASCE, 2005: 131 (6) ]

Transverse steel contribution to shear stress

Bar slip material model calibration

column

Bar-slip spring Mechanistic model :- envelope

Hysteretic model calibration

Strength deterioration model

-2 0 2 4 6 8 10 12 14 16-1000

-500

0

500

1000

slip (mm)

bar-

spri

ng fo

rce

(kN

)

Typical response envelope

Bar slip mechanistic model

Assumptions for anchorage response of bond within the joint region:

Bond stress uniform for elastic reinforcement, piecewise uniform for reinforcement

loaded beyond yield

Slip is the relative movement of reinforcement bar with respect to the joint perimeter

Slip is a function of strain distribution in the joint

Bar exhibits zero slip at zero bar stress

2

0

2fsl

fsE b Eslip s y

b b

ldd x dx f fA E E d

= =